# AP Calculus AB : Derivatives of functions

## Example Questions

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### Example Question #1 : Derivatives Of Functions

The Second Fundamental Theorem of Calculus (FTOC)

Consider the function equation (1)

(1)

The Second FTOC states that if:

1.  is continuous on an open interval
2.   is in .
3. and   is the anti derivative of

then we must have,

(2)

Differentiate,

Explanation:

Differentiate:

Both terms must be differentiated using the chain rule. The second term will use a combination of the chain rule and the Second Fundamental Theorem of Calculus. To make the derivative of the second term easier to understand, define a new variable so that the limits of integration will have the form shown in Equation (1) in the pre-question text.

Let,

Therefore,

Now we can write the derivative using the chain rule as:

Let's calculate the derivative with respect to  in the second term using the Second FTOC and then convert back to

Therefore we have,

### Example Question #1 : Computation Of The Derivative

Find the derivative.

Explanation:

Use the power rule to find the derivative.

### Example Question #2 : Computation Of The Derivative

Find the derivative of the following:

None of the above

Explanation:

To take the derivative you need to bring the power down to the front of the equation, multiplying it by the coefficient and then drop the power.

So:

becomes

because the degree of "x" is just one, and once you multiply 3 by 1 you get 3 and drop the power of "x" to 0. The second term is just a constant and the derivative of any term is just 0.

### Example Question #1 : Computation Of The Derivative

Find :

Explanation:

To do this problem you need to use the quotient rule. So you do

(low)(derivative of the high)-(high)(derivative of the low) all divided by the bottom term squared.

So:

Which, when simplified is:

### Example Question #1 : Derivatives Of Functions

Find :

Explanation:

This is the product rule, which is: (derivative of the first)(second)+(derivative of the second)(first)

So:

### Example Question #2 : Derivatives Of Functions

Find the derivative of the following:

Explanation:

This is a combination of chain rule and quotient rule.

So:

Which when simplified you get:

### Example Question #3 : Derivatives Of Functions

Find the derivative of the following:

Explanation:

This problem is just addition of derivatives using trigonometric functions.

So:

### Example Question #2 : Computation Of The Derivative

Find the derivative:

Explanation:

The is a quotient rule using a trigonometric function.

So:

You can pull out an "x" and cancel it to get:

### Example Question #4 : Derivatives Of Functions

Find the derivative:

Explanation:

This is the same concept as a normal derivative just with a negative in the exponent.

which becomes:

### Example Question #26 : Derivatives

Calculate :

Explanation:

This is a power rule that can utilize u-substitution.

So

where

So you get:

Plug "u" back in and you get:

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